What does this all mean? Time is relative. There is no absolute clock to say whether I am
right or you are right. All the observations I make in my reference frame will be totally
consistent, and so will yours.
We can express time dilation mathematically. If I were carrying a stopwatch and
measured a time interval, , you would get a different measure, t, for the amount of time
I had the stopwatch running.
The relation between these measures is:
So suppose I am moving at one-half the speed of light relative to you. If I measure 10
seconds on my stopwatch, you will measure the same time interval to be:
This equation has noticeable effects only at near light speeds. The difference between t
and is only a factor of. This factor—which comes up so frequently in
special relativity that it has been given its own symbol, —is very close to 1 unless v is a
significant fraction of c. You don’t observe things on a train moving at a slower rate, since
even on the fastest trains in the world, time slows down by only about 0.00005%.
Time Dilation and Simultaneity
Normally, we would think that if two events occur at the same time, they occur at the
same time for all observers, regardless of where they are. However, because time can
speed up or slow down depending on your reference frame, two events that may appear
simultaneous to one observer may not appear simultaneous to another. In other words,
special relativity challenges the idea of absolute simultaneity of events.
EXAMPLE
A spaceship of alien sports enthusiasts passes by the Earth at a speed of 0.8c, watching the
final minute of a basketball game as they zoom by. Though the clock on Earth measures a
minute left of play, how long do the aliens think the game lasts?Because the Earth is moving at such a high speed relative to the alien spaceship, time
appears to move slower on Earth from the aliens’ vantage point. To be precise, a minute
of Earth time seems to last: